ASTR 3830: Problem Set 4
Due in class Friday March 2nd
1) A spiral galaxy has a rotation curve which rises linearly from zero at the center to 200
km/s at 5 kpc from the center. The rotation curve then remains flat at a constant value of
200 km/s out to 15 kpc, beyond which it can’t be measured.
(a) Make a plot of M(r), the total mass enclosed within radius r, as a function of radius.
Assume that the mass distribution in the galaxy is spherically symmetric. Note: 1 kpc =
3.086 x 1021 cm. Use grams for the y-axis on the plot.
(b) Assume that the form of the rotation curve is dominated by dark matter. Plot the
density of the dark matter in g cm-3 versus radius.
(c) If this model applied to our own Galaxy, what would be the estimated mass in dark
matter interior to the Earth’s orbit in the Solar System (i.e. within 1 AU of the Sun)?
2) Consider a star of radius R* a distance r away from a supermassive black hole with
mass MBH. The star has mean density ρ*.
(a) Calculate the difference in the gravitational force from the black hole at distance r and
distance (r+R*) – this is called the tidal force. By equating this to the force that gas on the
stellar surface feels due to the star’s own gravity, show that the tidal force will be able to
overcome the star’s own self-gravity and rip the star apart if:
!
"
*<CMBH
r3
where C is a numerical factor which you should determine but whose exact value is not
too important.
(b) Hence calculate the minimum distance that a Solar type star could approach the
supermassive black hole at the Galactic Center before being tidally destroyed. The mass
of the Galactic Center black hole is about 4 million Solar masses.
(c) Recalling that a black hole has a Schwarzschild radius given by 2GMBH/c2, find the
value of the maximum black hole mass for which tidal disruption of a Solar type star can
occur (above this mass the tidal disruption radius is inside the Schwarzschild radius, and
the black hole would be able to swallow stars whole).